Quotients of strongly proper posets, and related topics Sean Cox - - PowerPoint PPT Presentation

Quotients of strongly proper posets, and related topics

Sean Cox

Virginia Commonwealth University scox9@vcu.edu

Forcing and its Applications Retrospective Workshop, March 2015

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Joint work with John Krueger.

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A conjecture of Viale-Weiss

The principle ISP(ω2): introduced by Weiss follows from PFA (Viale-Weiss), and many consequences of PFA factor through ISP(ω2). Conjecture (Viale-Weiss): ISP(ω2) is consistent with large continuum (i.e. > ω2).

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A conjecture of Viale-Weiss

The principle ISP(ω2): introduced by Weiss follows from PFA (Viale-Weiss), and many consequences of PFA factor through ISP(ω2). Conjecture (Viale-Weiss): ISP(ω2) is consistent with large continuum (i.e. > ω2). Theorem (C.-Krueger 2014) Proved the conjecture of Viale-Weiss. Developed general theory of quotients of strongly proper forcings.

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Outline

1

Approximation property and guessing models

2

Strongly proper forcings and their quotients

3

an application: the Viale-Weiss conjecture

4

Specialized guessing models, and a question

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Approximation property

Definition (Hamkins) Let (W , W ′) be transitive models of set theory such that: W ⊂ W ′ µ is regular in W We say (W , W ′) has the µ-approximation property iff whenever:

1 X ∈ W ′; 2 X is a bounded subset of W ; 3 ∀z ∈ W |z|W < µ =

⇒ z ∩ X ∈ W then X ∈ W .

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Approximation property

Definition (Hamkins) Let (W , W ′) be transitive models of set theory such that: W ⊂ W ′ µ is regular in W We say (W , W ′) has the µ-approximation property iff whenever:

1 X ∈ W ′; 2 X is a bounded subset of W ; 3 ∀z ∈ W |z|W < µ =

⇒ z ∩ X ∈ W then X ∈ W . We will focus on the case µ = ω1 throughout this talk.

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The class Gω1

Definition (Viale-Weiss) M is ω1-guessing, denoted M ∈ Gω1, iff |M| = ω1 ⊂ M and (HM, V ) has the ω1-approximation property (where HM is transitive collapse of M). Definition (Viale-Weiss) ISP(ω2) is the statement: for all regular θ ≥ ω2: Gω1 ∩ Pω2(Hθ) is stationary

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The class Gω1

Definition (Viale-Weiss) M is ω1-guessing, denoted M ∈ Gω1, iff |M| = ω1 ⊂ M and (HM, V ) has the ω1-approximation property (where HM is transitive collapse of M). Definition (Viale-Weiss) ISP(ω2) is the statement: for all regular θ ≥ ω2: Gω1 ∩ Pω2(Hθ) is stationary Theorem (Viale-Weiss) The Proper Forcing Axiom (PFA) implies ISP(ω2).

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The class Gω1

Definition (Viale-Weiss) M is ω1-guessing, denoted M ∈ Gω1, iff |M| = ω1 ⊂ M and (HM, V ) has the ω1-approximation property (where HM is transitive collapse of M). Definition (Viale-Weiss) ISP(ω2) is the statement: for all regular θ ≥ ω2: Gω1 ∩ Pω2(Hθ) is stationary Theorem (Viale-Weiss) The Proper Forcing Axiom (PFA) implies ISP(ω2). Generalization of theorems of Baumgartner, Krueger

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Consequences of PFA that factor through ISP

TP(ω2) Every tree of height and size ω1 has at most ω1 many cofinal branches (in particular no Kurepa trees)

together with 2ω1 = ω2 this yields ♦+(S2

1) (Foreman-Magidor)

Failure of (θ) for all θ ≥ ω2 (Weiss; actually failure of weaker forms of square) SCH (Viale) IAω1 =∗ Unifω1 and stronger separations (Krueger) Laver Diamond at ω2 (Viale from PFA, Cox from ISP plus 2ω = ω2)

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Consequences of PFA that factor through ISP

TP(ω2) Every tree of height and size ω1 has at most ω1 many cofinal branches (in particular no Kurepa trees)

together with 2ω1 = ω2 this yields ♦+(S2

1) (Foreman-Magidor)

Failure of (θ) for all θ ≥ ω2 (Weiss; actually failure of weaker forms of square) SCH (Viale) IAω1 =∗ Unifω1 and stronger separations (Krueger) Laver Diamond at ω2 (Viale from PFA, Cox from ISP plus 2ω = ω2) Even more consequences of PFA factor through “specialized” ISP; more on that later.

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Example: ISP(ω2) implies TP(ω2)

Let T be a tree of height ω2 and width < ω2. By stationarity of Gω1 there is an M ∈ Gω1 such that M ≺ (Hω3, ∈, T). Let σ : HM → M ≺ Hω3 be inverse of collapsing map of M; let α := M ∩ ω2 = crit(σ) and TM := σ−1(T) Our goal is to prove that HM | = “TM has a cofinal branch”.

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Example: ISP(ω2) implies TP(ω2)

Let T be a tree of height ω2 and width < ω2. By stationarity of Gω1 there is an M ∈ Gω1 such that M ≺ (Hω3, ∈, T). Let σ : HM → M ≺ Hω3 be inverse of collapsing map of M; let α := M ∩ ω2 = crit(σ) and TM := σ−1(T) Our goal is to prove that HM | = “TM has a cofinal branch”. Since (HM, V ) has the ω1-approximation property, it suffices to find (in V ) a cofinal b through TM such that every proper initial segment of b is an element of HM. But since T is thin, then TM = T|α. Pick any t on the α-th level of T; then t ↓ is a cofinal branch through TM = T|α and every proper initial segment is of course in HM.

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Outline

1

Approximation property and guessing models

2

Strongly proper forcings and their quotients

3

an application: the Viale-Weiss conjecture

4

Specialized guessing models, and a question

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Review of forcing quotients

A suborder P of Q is regular iff maximal antichains in P remain maximal antichains in Q.

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Review of forcing quotients

A suborder P of Q is regular iff maximal antichains in P remain maximal antichains in Q. Definition Suppose P is a regular suborder of Q and GP is P-generic. In V [GP] the (possibly nonseparative) quotient Q/GP is the set of q ∈ Q which are compatible with every member of GP. Order is inherited from Q. Q ∼ P ∗ ˇ Q/ ˙ GP

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Review of forcing quotients

A suborder P of Q is regular iff maximal antichains in P remain maximal antichains in Q. Definition Suppose P is a regular suborder of Q and GP is P-generic. In V [GP] the (possibly nonseparative) quotient Q/GP is the set of q ∈ Q which are compatible with every member of GP. Order is inherited from Q. Q ∼ P ∗ ˇ Q/ ˙ GP Important variation: “P is regular in Q below q”

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Strongly proper forcing

The following notion is due to Mitchell. Definition Given a poset P and a model M, a condition p ∈ P is an (M, P) strong master condition iff “M ∩ P is a regular suborder of P below p”. (we focus only on countable M)

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Strongly proper forcing

The following notion is due to Mitchell. Definition Given a poset P and a model M, a condition p ∈ P is an (M, P) strong master condition iff “M ∩ P is a regular suborder of P below p”. (we focus only on countable M) “P is strongly proper”: defined similarly to properness, using strong master condition instead of master condition.

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Examples and properties of strongly proper forcings

Examples: Todorcevic’s finite ∈-collapse Baumgartner’s adding a club with finite conditions adding any number of Cohen reals Various (pure) side condition posets of Mitchell, Friedman, Neeman, Krueger, and others.

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Examples and properties of strongly proper forcings

Examples: Todorcevic’s finite ∈-collapse Baumgartner’s adding a club with finite conditions adding any number of Cohen reals Various (pure) side condition posets of Mitchell, Friedman, Neeman, Krueger, and others. Key properties (Mitchell): absorbs Add(ω) (V , V P) has the ω1-approximation property

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Examples and properties of strongly proper forcings

Examples: Todorcevic’s finite ∈-collapse Baumgartner’s adding a club with finite conditions adding any number of Cohen reals Various (pure) side condition posets of Mitchell, Friedman, Neeman, Krueger, and others. Key properties (Mitchell): absorbs Add(ω) (V , V P) has the ω1-approximation property Remark: To get ω1 approx, suffices to be strongly proper wrt stationarily many countable models.

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Sketch of ω1-approx property from strong properness

Suppose 1P forces that ˙ b is a new subset of θ and that z ∩ ˙ b ∈ V for every V -countable set z. Let M ≺ (Hθ+, ∈, ˙ b, . . . ) be countable and let p be a strong master condition for M. Since M is countable then by assumption ˇ M ∩ ˙ b is forced to be in the ground

• model. Let p′ ≤ p decide this value.

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Sketch of ω1-approx property from strong properness

Suppose 1P forces that ˙ b is a new subset of θ and that z ∩ ˙ b ∈ V for every V -countable set z. Let M ≺ (Hθ+, ∈, ˙ b, . . . ) be countable and let p be a strong master condition for M. Since M is countable then by assumption ˇ M ∩ ˙ b is forced to be in the ground

• model. Let p′ ≤ p decide this value.

Let p′|M be a reduct of p′ into M ∩ P. Since ˙ b is forced to be new and ˙ b, p′|M ∈ M, then there are r, s ∈ M below p′|M which disagree about some member of M being an element of ˙

• b. Then

clearly they cannot both be compatible with a condition which decides ˇ M ∩ ˙

• b. In particular they cannot both be compatible with

p′. Contradiction.

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Quotients of strongly proper forcings

Question Suppose Q is strongly proper and P is a regular suborder. When does the quotient Q/ ˙ GP have the following properties? strongly proper “wrt V models”? ω1-approximation property?

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Quotients of strongly proper forcings

Question Suppose Q is strongly proper and P is a regular suborder. When does the quotient Q/ ˙ GP have the following properties? strongly proper “wrt V models”? ω1-approximation property? Remark: There are well-known examples of quotients of proper forcings that aren’t proper.

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The star condition

From now on we only deal with “well-met” posets: if p q then they have a GLB

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The star condition

From now on we only deal with “well-met” posets: if p q then they have a GLB Definition (Krueger) Assume P is a suborder of Q. ⋆(P, Q) denotes the statement: whenever p ∈ P and q1, q2 ∈ Q and p, q1, q2 are pairwise compatible, then there is a lower bound for all three. ⋆(Q) is the stronger statement that ⋆(Q, Q) holds. Examples where ⋆(Q) holds: Col(µ, θ) Todorcevic’s ∈-collapse Krueger’s adequate set forcing

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Key properties of ⋆(P, Q)

Lemma Assume ⋆(P, Q) and let GP be generic for P. Then in V [GP]:

• ∀q1, q2 ∈ Q/GP

q1 Q q2 = ⇒ q1 Q/ ˙

GP q2

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Key properties of ⋆(P, Q)

Lemma Assume ⋆(P, Q) and let GP be generic for P. Then in V [GP]:

• ∀q1, q2 ∈ Q/GP

q1 Q q2 = ⇒ q1 Q/ ˙

GP q2

• Proof: let q1, q2 ∈ Q/GP and suppose q1 ∧ q2 = 0 in Q; we will

prove that q1 ∧ q2 ∈ Q/GP, i.e. that q1 ∧ q2 is compatible with every member of GP. Let p ∈ GP. Then q1 ∧ p = 0 = q2 ∧ p. By ⋆(P, Q) we have q1 ∧ q2 ∧ p = 0.

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⋆(P, Q) implies strong master conditions survive in the quotient

Lemma Suppose ⋆(P, Q) holds and q is (M, Q) strong master condition. Then P ˇ q ∈ Q/ ˙ GP = ⇒ ˇ q is (M[ ˙ GP], Q/ ˙ GP) s.m.c.

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⋆(P, Q) implies strong master conditions survive in the quotient

Lemma Suppose ⋆(P, Q) holds and q is (M, Q) strong master condition. Then P ˇ q ∈ Q/ ˙ GP = ⇒ ˇ q is (M[ ˙ GP], Q/ ˙ GP) s.m.c. Proof sketch: Suppose p ∈ P forces that ˇ q ∈ Q/ ˙ GP (i.e. ˇ q ˙ GP). Then p must force that M[ ˙ GP] ∩ V = M; otherwise there is some p′ ≤ p forcing M M[ ˙ GP] ∩ V , but p′ still forces ˇ q ∈ Q/ ˙

• GP. So

let GP ∗ H be generic (in the 2-step iteration) with (p′, q) ∈ GP ∗ H. But q is in particular an (M, Q) master condition, so M = M[GP ∗ H] ∩ V ⊃ M[GP] ∩ V . Contradiction.

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Recall q is (M, Q) strong master condition, and we showed that if q ∈ Q/GP then in particular Q ∩ M = Q ∩ M[GP] =: QM. Now QM is regular in Q below q (this is Σ0 statement). Suppose q′ ≤ q, where q′ ∈ Q/GP. Let q′|M be a reduct of q′ into

• QM. We need to see that:

q′|M GP; this is straightforward, especially if q′|M ≥ q′ as is usually the case; and any extension of q′|M in QM/GP is compatible with q′ in Q/GP. Suppose q′′ is such a condition; so q′′ GP and is Q- compatible with q′. By the previous lemma (using the ⋆(P, Q) assumption), q′ and q′′ are compatible in Q/GP.

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A sufficient condition

Theorem (C.-Krueger) Suppose: Q is well-met; There is a stationary set S of countable models M for which Q has universal strong master conditions; P is a regular suborder of Q (possibly “below a condition”) ⋆(P, Q) holds Then P forces that Q/ ˙ GP is strongly proper for the stationary set

• f models of the form M[ ˙

GP] where M ∈ S. In particular, the quotient has the ω1 approximation property.

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A sufficient condition

Theorem (C.-Krueger) Suppose: Q is well-met; There is a stationary set S of countable models M for which Q has universal strong master conditions; P is a regular suborder of Q (possibly “below a condition”) ⋆(P, Q) holds Then P forces that Q/ ˙ GP is strongly proper for the stationary set

• f models of the form M[ ˙

GP] where M ∈ S. In particular, the quotient has the ω1 approximation property. REMARK: universality isn’t needed if you only want ω1-approx property.

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A counterexample

Quotients of strongly proper posets may fail to have the ω1-approximation property: Theorem (Krueger) Assume 2ω = ω1 and 2ω1 = ω2. Let Q be the forcing with coherent adequate sets of countable submodels of Hω3. Then Q has the following properties: Q is strongly proper and ω2-cc; Q forces CH Q adds a Kurepa tree on ω1 with ω3 many cofinal branches There is a regular suborder P of size ω2 such that P Q/ ˙ GP fails to have the ω1 approximation property

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Outline

1

Approximation property and guessing models

2

Strongly proper forcings and their quotients

3

an application: the Viale-Weiss conjecture

4

Specialized guessing models, and a question

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ISP and large continuum

Recall Viale-Weiss: proved PFA implies ISP(ω2); conjectured that ISP(ω2) is consistent with large continuum.

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ISP and large continuum

Recall Viale-Weiss: proved PFA implies ISP(ω2); conjectured that ISP(ω2) is consistent with large continuum. Theorem (C.-Krueger) Assume κ is a supercompact cardinal and θ ≥ κ arbitrary. Let: P be “adequate set forcing” to turn κ into ℵ2; (or Neeman’s side condition forcing; or Friedman’s; ...) Q = Add(ω, θ) Then V P×Q | = ISP(ω2) and 2ω = θ.

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Proof outline

Let G × H be generic for P × Q. Let θ ≥ ω2 = κ be regular and A = (Hθ[G × H], ∈, . . . ) be an algebra.

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Proof outline

Let G × H be generic for P × Q. Let θ ≥ ω2 = κ be regular and A = (Hθ[G × H], ∈, . . . ) be an algebra. Back in V let j : V → N be sufficiently supercompact with crit(j) = κ so that j[Hθ] ∈ N. P × Q is κ-cc and crit(j) = κ, so j : P × Q → j(P × Q) is a regular embedding; so we can force with the quotient j(P × Q)/j[G × H] (1) and lift j to j : V [G × H] → N[G ′ × H′]

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Proof outline

Let G × H be generic for P × Q. Let θ ≥ ω2 = κ be regular and A = (Hθ[G × H], ∈, . . . ) be an algebra. Back in V let j : V → N be sufficiently supercompact with crit(j) = κ so that j[Hθ] ∈ N. P × Q is κ-cc and crit(j) = κ, so j : P × Q → j(P × Q) is a regular embedding; so we can force with the quotient j(P × Q)/j[G × H] (1) and lift j to j : V [G × H] → N[G ′ × H′] N believes that j(P × Q) is strongly proper and the pair j[P × Q], j(P × Q) satisfies the star property. So N

• j[G × H]
• believes that the

quotient in (1) has the ω1-approximation property; so (HV

θ [G × H], N[G ′ × H′]) has ω1-a.p., and also

j

• HV

θ [G × H]

• ≺ j(A). Then use elementarity of j.

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Outline

1

Approximation property and guessing models

2

Strongly proper forcings and their quotients

3

an application: the Viale-Weiss conjecture

4

Specialized guessing models, and a question

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What Viale-Weiss really proved

Definition Let’s call M a specialized ω1 guessing model, and write M ∈ sGω1, iff a certain tree related to M is specialized; in particular M ∈ Gω1 and remains so in any outer model with the same ω1. They proved that under PFA, sGω1 ∩ Pω2(Hθ) (∩ICω1) is stationary for all θ ≥ ω2.

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Consequences of PFA which factor through specialized guessing models

If T is a tree of height and size ω1 then forcing with T collapses ω1 (Baumgartner) (together with assumption 2ω = ω2) Every forcing which adds a new subset of ω1 either adds a real or collapses ω2 (Todorcevic)

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Sketch of proof

In V consider the stationary set S := sGω1 ∩ Pω2(Hω2). Using stationarity of S and the assumption that 2ω = ω2, fix a ⊂-increasing (non-continuous) chain Mα | α < ω2 of elements of S whose union contains Hω1.

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Sketch of proof

In V consider the stationary set S := sGω1 ∩ Pω2(Hω2). Using stationarity of S and the assumption that 2ω = ω2, fix a ⊂-increasing (non-continuous) chain Mα | α < ω2 of elements of S whose union contains Hω1. Suppose W is an outer model of V which adds a new subset b of ω1, and doesn’t add a real. Then it doesn’t add new subsets of countable ordinals either, so for all ξ < ω1 we have b ∩ ξ ∈ HV

ω1 ⊂

• α<ω2

Mα

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Sketch of proof

In V consider the stationary set S := sGω1 ∩ Pω2(Hω2). Using stationarity of S and the assumption that 2ω = ω2, fix a ⊂-increasing (non-continuous) chain Mα | α < ω2 of elements of S whose union contains Hω1. Suppose W is an outer model of V which adds a new subset b of ω1, and doesn’t add a real. Then it doesn’t add new subsets of countable ordinals either, so for all ξ < ω1 we have b ∩ ξ ∈ HV

ω1 ⊂

• α<ω2

Mα In W define a function f : ω1 → ωV

2 by sending ξ to the least α

such that b ∩ ξ ∈ Mα. This is a cofinal map from ω1 → ωV

2 since

for any α < ω2, since b / ∈ Mα and Mα is G W

ω1 then there is some

ξ < ω1 such that b ∩ ξ / ∈ Mα.

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A new question

Our model of ISP(ω2) plus large continuum is NOT a model of the “specialized” version (because it has a tree of height and size ω1 whose forcing doesn’t collapse ω1). This suggests a natural modification of the Viale-Weiss question: Question Assume “specialized” ISP(ω2); i.e. suppose sGω1 is stationary for all Pω2(Hθ). Does this imply 2ω = ω2?

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